The Extended-Track Event Reconstruction for

MiniBooNE

R. B. Patterson 1 b, E. M. Laird b, Y. Liu a, P. D. Meyers b,I. Stancu a, H. A. Tanaka 2 b

aUniversity of Alabama, Department of Physics and Astronomy, Tuscaloosa, AL,35487 USA

bPrinceton University, Department of Physics, Joseph Henry Laboratories,Princeton, NJ 08544 USA

Abstract

The Booster Neutrino Experiment (MiniBooNE) searches for νµ → νe oscillationsusing the ∼1 GeV neutrino beam produced by the FNAL Booster synchrotron.The array of photomultiplier tubes (PMTs) lining the MiniBooNE detector recordsCherenkov and scintillation photons from the charged particles produced in neu-trino interactions. We describe a maximum likelihood fitting algorithm used toreconstruct the basic properties (position, direction, energy) of these particles fromthe charges and times measured by the PMTs. The likelihoods returned from fit-ting an event to different particle hypotheses are used to categorize it as a signal νeevent or as one of the background νµ processes, in particular charged current quasi-elastic scattering and neutral current π0 production. The reconstruction and eventselection techniques described here can be applied to current and future neutrinoexperiments using similar Cherenkov-based detection.

Key words:PACS:

1 Now at the California Institute of Technology, Physics Department 103-33,Pasadena, CA 91125, USA.

2 Now at the University of British Columbia, Department of Physics and Astron-omy, Vancouver, BC V6T 1Z1, Canada.

Preprint submitted to Elsevier 1 October 2018

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1 Introduction

Cherenkov detectors of &1 kiloton have played a key role in the estab-lishment of the phenomenon of neutrino oscillations [1a–b,2a–c,3a–c]. Thesedetectors typically consist of a large volume of a homogeneous transparentmedium (water or mineral oil) with a high index of refraction surrounded byan array of photomultiplier tubes (PMTs). Cherenkov photons produced bycharged particles emerging from the neutrino interactions and traversing themedium with β > 1/n (where n is the index of refraction and β = v/c) aredetected by the PMTs. The photons are emitted at an angle θC relative to thetrack direction, where cos θC = 1/nβ. The radiation is azimuthally symmetricabout the track direction, resulting in a ring-like pattern that can be identi-fied on the PMT array. The quantity, spatial distribution, and arrival timesof these photons provide information on the location, direction, and energy ofthe particle. We refer to the extraction of such information from the chargeand time measurements of the PMTs as “reconstruction.” An analysis of thering profile can also provide information on the identity of the particle and, ifmultiple rings are detected, the number of particles.

In this paper, we discuss the event reconstruction algorithms used in theBooster Neutrino Experiment (MiniBooNE), which searches for an excess ofνe interactions indicative of νµ → νe oscillations using the Fermilab ∼1 GeVneutrino beam. The MiniBooNE detector is a sphere of radius 610.6 cm filledwith Marcol 7 mineral oil (n ≈ 1.47) and divided into two optically isolatedregions by an opaque shell of radius 575 cm, concentric with the sphere. Thesurface of the inner “main” region is instrumented with an array of 1280inward-facing 8 in. PMTs which detect the light produced by the neutrinointeractions. The outer “veto” region is instrumented with 240 PMTs and isused to tag charged particles that enter the detector from outside (e.g., cosmicmuons) or that exit the main region. Though no scintillator was added to theMarcol 7, it scintillates weakly, resulting in the production of delayed isotropiclight for particles with sub- and super-Cherenkov velocities. A more detaileddescription of the experiment can be found in Refs. [4] and [5].

2 Neutrino Interactions at MiniBooNE

The ∼1 GeV neutrino beam at MiniBooNE results in interactions withrelatively low outgoing particle multiplicity. The largest interaction channel isthe charged current quasi-elastic (CCQE) process:

ν` + n→ `− + p (1)

2

which accounts for ∼40% of all neutrino interactions in the MiniBooNE de-tector. Since the recoil proton of Eq. (1) is typically below Cherenkov thresh-old, only the outgoing lepton produces significant light. In modeling suchevents within a reconstruction algorithm, one can therefore consider only theCherenkov and scintillation light produced by the outgoing lepton. While therecoiling nucleon can produce scintillation light, this additional source of lightis not considered in the reconstruction.

A muon in the MiniBooNE detector exhibits minimum-ionizing behav-ior through most of its path with little chance of radiative energy loss. Incontrast, electrons typically induce electromagnetic showers, resulting in ad-ditional electrons and positrons that emit Cherenkov light. Thus, muons andelectrons have significantly different Cherenkov ring patterns. This differenceis the basis for distinguishing a muon track from an electron track, and thereconstruction algorithm has a model for each.

The next most abundant process in MiniBooNE is single pion production,which occurs primarily via ∆ resonance or coherent scattering:

ν` +N → `− +N ′ + π (CC π)

ν` +N → ν` +N ′ + π (NC π)(2)

where N (′) denotes a nucleon in the case of resonance production and a nucleusin the case of coherent production. Of particular interest is neutral current(NC) π0 production. The two photons from π0 → γγ decay induce electro-magnetic showers indistinguishable in MiniBooNE from those induced by elec-trons. Since the recoil nucleon/nucleus is typically below Cherenkov threshold,these NC π0 events are well-described in the reconstruction algorithm by twoelectron-like tracks pointing back to a common vertex. The presence of twodistinct Cherenkov rings allows for the separation of π0 events from electron(νe CCQE) events. However, π0 misidentification (for example, due to a largeenergy asymmetry between the two photons or due to a small photon open-ing angle which leaves the Cherenkov rings overlapping) accounts for most ofthe reducible background in the νµ → νe oscillation search. Thus, successfulreconstruction and identification of π0 events is critical.

While other channels can be accommodated by the reconstruction algo-rithm, the νe appearance oscillation analysis uses only four event models: singleelectron track, single muon track, two γ tracks, and two γ tracks with a π0

invariant mass.

3

3 The Detector Response

The Marcol 7 mineral oil used in MiniBooNE exhibits a rich array of opti-cal phenomena despite its ∼20 m extinction length near the peak of PMTsensitivity (∼400 nm). Cherenkov and scintillation light production is ac-companied by photon absorption, fluorescence (with several possible excita-tion/emission spectra and lifetimes), and Rayleigh and Raman scattering. Theleft plot in Figure 1 summarizes the rates of various processes as a function ofwavelength. The cumulative extinction rate is shown as the black line. In thenear-ultraviolet region (<320 nm), fluorescence processes dominate the extinc-tion rate. The fluorescence lifetimes range from 1−35 ns. In the visible region(>320 nm), the dominant processes are Rayleigh scattering and absorption.

Fig. 1. Left: Rates of optical processes in Marcol 7 as a function of wavelength.The solid black line is the overall extinction rate obtained from laboratory measure-ments. The dashed black line is the extrapolated rate based on in situ data. Thecurves labeled “Fluor n” are the excitation rates for the four identified fluorescenceprocesses. The light blue points represent the measured rates of Rayleigh scatter-ing, and the dashed light blue and gray lines represent extrapolated rates. Right:Reconstructed photon arrival times for R1408 (top) and R5912 (bottom) PMTs for397 nm light flashed from the center of the detector. The black histogram is thedistribution from data and the blue is the complete Monte Carlo simulation. Thegreen (red) shows the simulation with reflections (and scattering) turned off.

The 1520 8 in. PMTs in MiniBooNE are of two types: 322 model R5912PMTs and 1198 model R1408 PMTs, both from Hamamatsu. All of the R5912PMTs are located in the main PMT array. The PMT time response (particu-larly the late-pulsing behavior) and the variation of PMT efficiency with inci-dent angle have been characterized in external measurements using a pulsedLED [6]. The PMTs in situ have been studied using a laser calibration systemthat produces sub-nanosecond bursts of nearly isotropic light in the detector.

4

The right panel of Figure 1 shows the times of PMT hits recorded in lasercalibration events relative to their expected times. The colored histogramscome from Monte Carlo simulation, with the red curve showing PMT andelectronics timing effects (including pre- and late-pulsing), the green curve in-cluding photon scattering, and the blue curve adding photon reflections fromthe PMTs and the surface of the main detector region. The resulting timestructure matches well with that seen in data (black points). We note thatonly the earliest photoelectron’s time is reported when multiple photoelec-trons are present in a hit. (A fresh hit can be seen after an electronics deadtime of ∼300 ns.) The reported charge, however, does account for additionalphotoelectrons.

Absorption, scattering, reflections, and fluorescence processes influenceboth the topology and the time structure of hits. Additionally, scintillationlight, emitted with a lifetime of ∼35 ns, adds intrinsically delayed light tothe prompt Cherenkov photons. The event models within the reconstructionalgorithm account for all these phenomena. In the discussion, we refer toCherenkov and scintillation light that travels undisturbed from the particletrack to the PMT as “direct” light. Light that undergoes fluorescence, scat-tering, or reflection we term “indirect” light.

A detailed GEANT3-based [7] Monte Carlo simulation of the MiniBooNEdetector serves as the central tool for developing the reconstruction algorithm’spredictive models. The accuracy of the models influences performance throughresolution and biases in extracted track parameters and through the particleseparation capabilities of the likelihood-based hypothesis testing. To balancethe need for likelihoods that are as accurate as possible with the desire tocontain the computational demands, detector and track model informationextracted from the Monte Carlo simulation is tabulated, approximated, orparametrized wherever possible while minimizing any sacrifice in performance.Also, thanks to the excellent stability of the detector’s optical properties, asingle configuration of the reconstruction is sufficient for use on all MiniBooNEphysics data.

4 The Single Track Model

The reconstruction of a single track in the detector, whether an electronor muon, is based on a model with seven parameters:

• the starting point: x0, y0, z0

• the starting time: t0• the direction: θ0, φ0

• the kinetic energy: E0.

5

The starting point and time are referred to as the “vertex” of the event. Thedirection is described in terms of the polar angle θ0 defined with respect to thedirection of the neutrino beam and the azimuthal angle φ0 about this beamaxis. For the electron model, the stochastic variations in the electromagneticshowers are modeled in average – no attempt is made to account for event-by-event variations in the shower profile. This is also true of the much smallervariations in the muon propagation (e.g., straggling). This simplification allowsthe track to be fully specified from its initial properties summarized in theseven parameters above. We refer to this vector of parameters as x.

The observables of an event are the measurements from the 1280 PMTsin the main region of the detector. For each PMT, we have:

• a bit indicating whether the tube registered a hit• if the tube was hit, the measured charge of the hit• if the tube was hit, the measured time of the hit.

Assuming that the PMTs behave independently, the likelihood for an observedset of PMT measurements given track parameters x can be expressed as:

L(x) =∏

unhit

(1− P (i hit; x))×∏hit

P (i hit; x) fq(qi; x) ft(ti; x) (3)

where the products are taken over all unhit and hit PMTs and:

• P (i hit; x) is the probability that the ith tube is hit, given parameters x.• qi, ti are the measured charge and time on the ith PMT.• fq(qi; x) is the probability distribution function (PDF) for the measured

charge on the ith PMT, given x, evaluated at qi.• ft(ti; x) is the PDF for the measured time given x, evaluated at ti.

It is convenient to work with the negative logarithm of L, and since the charge-and time-related portions of the likelihood are distinct, it is helpful to define:

F (x) ≡ − logL(x) ≡ Fq(x) + Ft(x) (4)

where

Fq(x) = −∑unhit

log(1− P (i hit; x)) −∑hit

log (P (i hit; x)fq(qi; x))

Ft(x) = −∑hit

log (ft(ti; x))(5)

For brevity, we refer to Fq and Ft as the charge and time likelihoods, respec-tively, although they are the negative logarithms of the likelihoods.

6

4.1 The Charge Likelihood

If the number of photoelectrons n produced in a PMT is known, then fq(q)for the observed charge q is fully specified in terms of the response propertiesof the PMT (assumed fixed) without any reference to any other property ofthe detector. Further, n is a Poisson variable whose mean µ is a function ofthe track parameters x. These considerations motivate a two-step approachto calculating the charge likelihood.

In the first step, using the known optical photon and particle propaga-tion properties of the detector, one determines for a given particle type (elec-tron/muon) and set of track parameters x the average number of photoelec-trons that a particular PMT should observe. This quantity is referred to as µi,the “predicted charge” for the ith PMT. In calculating the predicted charge,one considers the quantity and angular distribution of light produced by thetrack, the absorption, scattering, and fluorescence of light as it traverses themineral oil, the acceptance of the PMT, and anything else that influences themapping x 7→ µi.

For the second step, Fq given in Eq. (5) can be rewritten in terms of thepredicted mean charges µi:

Fq(x) ≡ −∑unhit

log(1− P (i hit;µi))−∑hit

log(P (i hit;µi)fq(qi;µi)) (6)

where P and fq are now functions of the µi, with the mapping x 7→ µi fromthe first step left implicit. These two functions depend only on the propertiesof the PMT response and are, in particular, independent of the happeningswithin the detector volume. This stepwise approach decouples the track andoptical photon model from the PMT response model. The latter is establishedonce using laser calibration data. The laser system provides a source of ap-proximately isotropic and prompt light with controllable intensity. The valueof µ for a particular laser intensity can be determined from the occupancy ofthe PMTs. The function fq(q;µ) for each PMT type can be determined byobtaining the distribution of measured charges q seen at various intensitiesµ. These distributions incorporate all processes associated with the chargeresponse, such as the cascade of charge down the dynode chain, the amplifi-cation and digitization of the anode signal, and the conversion of the digitalreadouts into calibrated charge values. Examples of fq(q;µ) obtained in thisway are shown in Figure 2.

7

00.10.20.30.40.50.60.7

1 2 3 4 5 6 7 8 9 10

Mean 1.376

q (PE)

f(q;µ

=0.7

) (P

E-1)

R1408 PMTs

00.10.20.30.40.50.60.7

1 2 3 4 5 6 7 8 9 10

Mean 1.452

q (PE)

f(q;µ

=0.7

) (P

E-1)

R5912 PMTs

Fig. 2. Examples of f(q;µ) for µ = 0.7 photoelectrons. Note that the distributiononly includes cases where the PMT registers a hit.

4.2 Calculating the Predicted Charge µ

We begin our description of the predicted charge calculation with a sim-ple scenario involving a fictitious point source of isotropic direct light. This isthen generalized to a line source of isotropic light, appropriate for the scin-tillation production of a track. We then consider a line source in which theemission has a non-trivial (but azimuthally symmetric) angular distribution,representative of Cherenkov radiation. Finally, we account for the contributionof indirect light (i.e., scattering, reflections, etc.) from both the scintillationand the Cherenkov emission. In order to simplify notation and without loss ofgenerality, the discussion involves fixed but arbitrary PMT location and trackparameters x.

One needs only two parameters to describe the geometric relationshipbetween an isotropic point source and a PMT. We choose (1) the distancer between the source and the PMT and (2) the angle of incidence η at thePMT, with η = 0 corresponding to incidence parallel to the PMT axis. The

8

s

(s)

r(s)

extended track

s

(s)

r(s)

(s)

Fig. 3. Geometry of a line source of isotropic scintillation light (left) and directionalCherenkov light (right). In each case s is the distance from the origin of the track,r(s) is the distance to the PMT from the point along the track, and η(s) is the angleof PMT incidence for light emitted from this point. The presence of Cherenkov lightrequires the additional parameter θ(s) describing the light emission angle.

predicted charge on the PMT can be written

µpoint,sci = Φsci Ω(r) Tsci(r) ε(η) (7)

where Φsci is an event-energy-dependent scintillation light yield (i.e., Φsci =Φsci(E0) ), Ω(r) is a r-dependent solid angle factor, Tsci(r) is the transmissionof the oil and PMT glass as a function of light propagation distance, and ε(η)is the acceptance of the PMT as a function of the angle of incidence. Thetransmission depends on the wavelength spectrum of the light source and,thus, is specific in this case to scintillation. The wavelength-dependent PMTphotocathode efficiency is also included in T (r). The solid angle factor andPMT angular acceptance are purely geometric, independent of the particularsof the light production.

To generalize to an extended source of scintillation light, as depicted inFigure 3(left), the latter three factors in Eq. (7) become functions of s, thedistance along the track from its origin. We must also include an emissionprofile ρsci(s) describing the distribution of scintillation production along thetrack. This profile satisfies

∫∞−∞ ρ(s)ds = 1. The predicted charge can now be

obtained by integrating along the track’s axis:

µsci = Φsci

∞∫−∞

ds ρsci(s) Ω(s)Tsci(s) ε(s) . (8)

The dependences of Ω, Tsci, and ε on r and η have been recast as dependenceson s, as the relationship s 7→ (r, η) is fixed for given track parameters andPMT location. Figures 4 and 5 show ρsci(s) for 300 MeV muons and electrons,obtained via Monte Carlo simulation. The distribution is relatively flat formuons until the end of the track, where the ionization rate per unit track length

9

distance along track, s (cm)

0

0.002

0.004

0.006

0.008

0.01

0.012

-50 0 50 100 150 200 250

(s

) (c

m-1)

300 MeV muon

Fig. 4. Scintillation emission profile for 300 MeV muons as a function of s, thedistance along the track.

distance along track, s (cm)

00.0010.0020.0030.0040.0050.0060.0070.0080.009

-50 0 50 100 150 200 250

(s

) (c

m-1)

300 MeV electron

Fig. 5. Scintillation emission profile for 300 MeV electrons as a function of s, thedistance along the track.

becomes larger. (Saturation effects are taken into account in the simulation.)For electrons, the distribution reflects the showering behavior of the track,with ρsci(s) rising and falling with the number of shower particles.

We now consider an extended track emitting light with a non-trivial an-gular distribution, as is the case for Cherenkov radiation. We introduce θ, theangle to the PMT from the track, as shown in Figure 3(right), and expressthe predicted charge µCh due to Cherenkov radiation as:

µCh = ΦCh

∞∫−∞

dsρCh(s) Ω(s)TCh(s) ε(s) g(cos θ(s); s) (9)

This expression differs from its scintillation counterpart by the presenceof an angular emission profile g(cos θ(s); s). Note that g depends on s in twodistinct ways: the angular profile of the light changes as the track propagatesand loses energy, and the angle θ to the PMT changes depending on which

10

part of the track we are considering.

Figures 6 and 7 show ρCh(s) and g(cos θ; s) for simulated 300 MeV muonsand electrons. As the muon propagates, loses energy, and approaches theCherenkov threshold, the rate of Cherenkov radiation per unit track lengthdecreases and the Cherenkov angle becomes smaller. The scattering of themuon, which causes deviations of the track from its original direction, hasbeen included in the Monte Carlo simulation and results in the spread of theangular distribution about the nominal cos θC . For electrons, ρCh(s) followsthe shower profile, like ρsci(s). The presence of the shower particles is readilyapparent in the g(cos θ; s) distribution, which becomes substantially wider asthe “track” propagates.

In writing Eq. (9), a few simplifications are made. A more rigorous treat-

ment would involve d2P (cos θ,φ)d cos θdφ

, the differential probability of sending an emit-

ted Cherenkov photon in the direction (cos θ, φ), which would be integrated

over the solid angle subtended by the PMT. By assuming that d2P (cos θ,φ)d cos θdφ

is con-stant over the PMT face, the integration can be effected by simply multiplyingthe differential probability by the solid angle Ω. Also, the azimuthal symme-try of the emission reduces the two-dimensional PDF to a one-dimensionalexpression, namely g(cos θ). We take g(cos θ) to satisfy

1∫−1

d cos θ g(cos θ; s) = 1 (10)

for all values of s, with the result that a factor of (4π)−1 is absorbed into thedefinition ΦCh.

4.3 Indirect Light (scattering, fluorescence, reflections)

The above formalism determines the predicted charge for light arriving atthe PMTs directly from the track without any redirection. However, the de-tector has sources of indirect light from scattering, fluorescence, and reflectionas discussed in Section 3.

The geometry for indirect light given scintillation (isotropic) emission isshown in Figure 8(left), where an infinitesimal element ds along the track issituated at radius R from the center of the detector and at angle Θ relativeto the position of the PMT. The direct light from this track element is simplythe integrand of Eq. (8):

dµdirectsci = ds Φsci ρsci(s) Ω(s)Tsci(s) ε(s) . (11)

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00.0010.0020.0030.0040.0050.0060.0070.0080.009

-50 0 50 100 150 200 250

0.2 0.4 0.6 0.8 10

50100

05

10152025

0

20

40

60

80

100

120

0.2 0.4 0.6 0.8 10

5

10

15

20

25

C

h(s)

(cm

-1)

distance along track, s (cm)

cos cos

g(co

s; s

)

g(co

s; s

)

s (c

m)

s (cm

)

300 MeV muon

Fig. 6. Top: Cherenkov emission profile for 300 MeV muons as a function of s, thedistance along the track. Bottom: Two-dimensional emission profile showing theangular distribution of Cherenkov light as a function of s for a 300 MeV muon.

00.0010.0020.0030.0040.0050.0060.0070.0080.009

-50 0 50 100 150 200 250

0.2 0.4 0.6 0.8 10

50100

2468

101214

0

20

40

60

80

100

120

0.2 0.4 0.6 0.8 1

2

4

6

8

10

12

14

C

h(s)

(cm

-1)

distance along track, s (cm)

cos cos

g(co

s; s

)

g(co

s; s

)

s (c

m)

s (cm

)

300 MeV electron

Fig. 7. Top: Cherenkov emission profile for 300 MeV electrons as a function of s,the distance along the track. Bottom: Two-dimensional emission profile showing theangular distribution of Cherenkov light as a function of s for a 300 MeV electron.

12

ds

R

Θ

tank center

ds

R

Θ

track points a bit out of the plane

Fig. 8. Geometry of indirect light from an isotropic (left) and directional (right)source of light.

An analytic expression for the indirect light would involve an elaborate inte-gral over emission angles and scattering points throughout the tank. Ratherthan attempt this, we observe that the value of such an integral must beproportional to the source strength and must otherwise depend only on thetopological variables R and Θ. The source strength dependence can be elimi-nated by forming a ratio of the indirect and direct light predictions:

Asci(R, cos Θ) ≡ dµindirectsci

dµdirectsci

. (12)

Asci, which we refer to as the scattering table and which we build via thedetector simulation, is a property only of the detector optics and the (R, cos Θ)of the track element. With this table, the indirect light contribution can beimmediately incorporated into the expression for predicted charge:

µsci = Φsci

∞∫−∞

ds ρsci(s) Ω(s)Tsci(s)ε(s) [1 + Asci (R(s), cos Θ(s))] (13)

where the dependence of R and Θ on s has been made explicit.

For Cherenkov light, the situation is more complex since the light emissionis anisotropic. Two additional variables are needed to specify the direction ofa vector source relative to the PMT position and the tank center. We makethe following non-unique choice for these two variables, as shown in Figure8(right):

• θ: the angle between (1) the source direction vector and (2) the source-to-PMT ray (the same θ as defined elsewhere).• φ: The angle between (1) the plane containing the tank center, the PMT,

and the source, and (2) the plane containing the track and the tank center.

The intensity of the indirect Cherenkov light is normalized with respect

13

to a fictitious isotropic Cherenkov source with predicted charge dµdirect,isoCh , so

that:

ACh(R, cos Θ, cos θ, φ) ≡ dµindirectCh

dµdirect,isoCh

(14)

The total Cherenkov contribution to the mean predicted charge, includingboth direct and indirect components is

µCh = µdirectCh + µindirect

Ch (15)

where µdirectCh is given by Eq. (9) and µindirect

Ch is given by

µindirectCh = ΦCh

∞∫−∞

ds [ρCh(s) Ω(s)TCh(s) ε(s)

× ACh (R(s), cos Θ(s), cos θ(s), φ(s))]

(16)

Implicit in this expression is the assumption that the Cherenkov angular emis-sion profile does not change as the track propagates (and loses energy). Thissimplification has negligible impact, as the processes that result in indirectlight tend to destroy the initial emission pattern anyway.

We now have a complete charge prediction, where the total predictedcharge for the PMT is given by:

µ = µCh + µsci (17)

4.4 Computation of Integrals

The predicted charge integrals above would be impractical to evaluate withsufficient spatial granularity within a maximum likelihood fit, where they mustbe evaluated multiple times for every PMT in every event. To mitigate this,all integrations are performed and tabulated beforehand as follows.

The integrand of Eq. (8), which pertains to direct scintillation light, is theproduct of a “source” factor Φsciρsci(s) and an “acceptance” factor

J(s) ≡ Ω(s)Tsci(s)ε(s) . (18)

As in Eq. (8), the dependence on r and η is recast through s. If J(s) varies

14

gradually enough, it can be approximated with a parabolic form:

J(s) = j0 + j1 s+ j2 s2 . (19)

An example of this is shown in Figure 9. The predicted charge then becomes

µdirectsci = Φsci

j0 ∞∫−∞

ds ρsci(s) + j1

∞∫−∞

ds ρsci(s) s+ j2

∞∫−∞

ds ρsci(s) s2

. (20)

The first integral is identically unity. The remaining two depend on the energyE0 via ρsci(s), but they depend on no other track parameters. This allows oneto tabulate the integrals beforehand as a function of E0, eliminating theirevaluation from the actual likelihood calculation.

The parabolic coefficients ji are obtained by evaluating Eq. (18) at threepoints along the track: the start of the track (s= 0), the midpoint of theCherenkov emission profile (s= ∆smid) and twice this distance from the start(s= 2∆smid). The choice of where to evaluate J(s) is somewhat arbitrary; onecould choose any three points that sample a large fraction of the range overwhich light is produced. The extension to indirect isotropic light is straight-forward with a redefinition of J(s) that incorporates the scattering table:

J indirectsci (s) ≡ Ω(s)Tsci(s) ε(s)Asci (R(s), cos Θ(s)) . (21)

An analogous three-integral expression can be formed for the Cherenkovpredicted charge:

µdirectCh = ΦCh

j0

∞∫−∞

ds ρCh(s)g(cos θ(s); s)

+ j1

∞∫−∞

ds ρCh(s) g(cos θ(s); s)s

+ j2

∞∫−∞

ds ρCh(s) g(cos θ(s); s)s2

.

(22)

As in the scintillation case, the integrals depend on the energy of the track.However, they also depend on two parameters which define the PMT-trackgeometry. We choose these parameters to be the vertex-to-PMT distance,r(0); and the cosine of the angle-to-PMT viewed from the vertex, cos θ(0).Labeling the integrals ICh

i , we obtain

µdirectCh = ΦCh

(j0ICh

0 + j1ICh1 + j2ICh

2

). (23)

15

3.48 m

s

1.19

m

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0 0.2 0.4 0.6 0.8 1 1.2

∆smid

full J(s)parabolic J(s)

s (m)

J(s)

(1

0-3)

Fig. 9. Example of J(s) for the track geometry shown in the top figure, wherea muon 3.48 meters from a PMT moves perpendicular to the PMT axis. J(s) isevaluated at three points located along the track, corresponding to the three pointsin the figure. In the bottom panel, the parabolic approximation for J(s) based onthe three evaluated points (dashed red) is compared to the exact form (solid black).

Although the integralsIChi

depend on three parameters, it is still feasible

to tabulate their values ahead of time, eliminating their explicit evaluationwithin the likelihood calculation.

For indirect Cherenkov light, recall that the scattering table ACh is nor-malized to a fictitious isotropic Cherenkov source. Thus, the parabolic method

16

used for indirect scintillation can be applied here to give

J indirectCh (s) ≡ Ω(s)TCh(s) ε(s) ACh (R(s), cos Θ(s)) , (24)

with two energy-dependent integrals IChi=1,2 =

∫dsρCh(s)si to be tabulated.

4.5 The Time Likelihood

The time portion of the likelihood involves the PDF ft(t; x) for the mea-sured PMT hit time t given track parameters x. Some of the dependence onthe track parameters can be eliminated by using the “corrected time” tc:

tc = t− t0 −r(∆smid(E0))

cn− ∆smid(E0)

c. (25)

The expression removes from t three terms, namely the starting time of thetrack t0, the expected time for light to propagate from the track midpointto the PMT (with the speed of light in mineral oil given by cn), and thetime for the particle to propagate from its starting point to the midpoint.The dependence of ∆smid on the track energy E0 is made explicit. Defining tc

with respect to the track midpoint (as opposed to, say, the start of the track)improves the validity of the simplifications we make below.

Given the t 7→ tc substitution, we seek the PDF ftc(tci ; x) for the corrected

time given arbitrary PMT-track configurations. The space of configurations isfive dimensional. However, producing tables of ftc(t

c) as a function of five pa-rameters is impractical. To reduce the task, we make the assumption that thecorrected time PDF depends only on the track energy, the predicted “prompt”charge, and the predicted “late” charge, where “prompt” corresponds roughlyto light arriving at the PMT directly from the track without delays induced byemission lifetimes, scattering, etc. Loosely, the assumption is that the shape ofthe corrected time spectrum is dominated by the physical extent of the track(characterized by its energy), and by the amount of prompt and late lightreaching the PMT. The extent of the track affects the spread of possible hittimes, since there is a spread of photon production times, while the amountsof prompt and late light affect the proportion of peaked “prompt” responseto the tail of “late” response.

This assumption reduces the configuration to three dimensions. We makeone further simplification by assuming that for a given energy E0, ftc(t

c) canbe modeled by separate primitive prompt and late distributions, indexed byµprompt and µlate, respectively, from which we can construct the full PDF. Asa result, the PDF is indexed not by the fundamentally two-dimensional space

17

)dezila

mro

n tin

u(

0

0.05

0.1

-20 -15 -10 -5 0 5 10 15 20

0.1589E-01/ 68

Constant 0.1339

Mean -0.2033

Sigma 1.761

0

0.05

0.1

0.15

-20 -15 -10 -5 0 5 10 15 20

0.2230E-01/ 85

Constant 0.1593

Mean -0.4301

Sigma 1.475

0

0.05

0.1

0.15

0.2

-20 -15 -10 -5 0 5 10 15 20

0.2968E-01/ 76

Constant 0.2042

Mean -1.240

Sigma 1.139

corrected time (ns)

0.046 < µCh

< 0.051

corrected time (ns)

0.45 < µCh

< 0.52

corrected time (ns)

4.4 < µCh

< 5.1

direct

direct

direct

Gaussian fit values:

Gaussian fit values:

Gaussian fit values:

Fig. 10. Distributions of tc for direct Cherenkov light for R5912 PMTs. The primarydifference between the two PMT types is that the R1480 PMTs show wider prompttiming peaks than do the newer R5912 PMTs.

of (µprompt, µlate), but by µprompt and µlate separately, with the full PDF cal-culated on-the-fly as described below. In practice, Cherenkov and scintillationprimitive distributions are created and used as proxies for the desired promptand late distributions, respectively.

The Cherenkov (prompt) primitive distributions are created by simulatingparticles throughout the detector with isotropically chosen directions and fixedenergy E0. The particles are created with direct Cherenkov light only; all othersources of light (scintillation, scattering, etc.) are turned off. For each event,the true track parameters are used to evaluate the direct Cherenkov predictedcharge µdirect

Ch for each hit. Histograms of the corrected time tc of the hits areproduced for various ranges of predicted charge. Figure 10 shows three such tc

distributions for 300 MeV muons. Since only direct Cherenkov light is present,the histograms show no late-time features. The shapes of the time spectra de-pend on the µdirect

Ch values involved, with the distributions becoming narrowerand earlier with increasing predicted charge due to the increasing probabil-ity that an early photon will be recorded at the PMT. The tc distribution ineach predicted charge range is fit to a Gaussian distribution, and the resultingGaussian parameters (mean and width) are subsequently parametrized across

18

-2-1.8-1.6-1.4-1.2

-1-0.8-0.6-0.4-0.2

0

10-1

1 10

0.80.9

11.11.21.31.41.51.61.71.8

10 -1 1 10

fitte

d m

ean

(ns)

Ch

(PE)

Ch

(PE)

fitte

d R

MS

(ns)

direct

direct

Fig. 11. For 300 MeV muons, the mean and width parameters from Gaussian fits likethose in Figure 10 versus µdirect

Ch . The dependence is parameterized as a sixth-orderpolynomial in log(µdirect

Ch ).

µdirectCh using a sixth-order polynomial in log(µdirect

Ch ). Figure 11 shows an exam-ple of these “second-level” fits for 300 MeV muons. The procedure is repeatedat many values of E0, with the two second-level fits providing seven parame-ters for the Gaussian mean and seven parameters for the Gaussian width ateach energy. The energy dependences of these 14 parameters are then fit asa function of E0 to fourth-order polynomials in a third-level parametrization,as shown in Figure 12 for the first seven of the second-level parameters. Inaddition to conveniently summarizing the dependence of the time response onµdirect

Ch and E0, the parametrizations provide a smooth likelihood surface, asrequired by the minimization algorithm.

This completes the prompt primitive distributionsGCh(tc;E0, µdirectCh ). They

are calculated for a given predicted charge µdirectCh at a PMT for a track with

energy E0 by evaluating the 14 third-level curves which parametrize the E0

dependence of the second-level functions. These 14 values then determine thetwo functions which parametrize the dependence of the mean and width of the

19

02468

500 1000 1500 2000

00.5

11.5

500 1000 1500 2000

0

0.5

1

500 1000 1500 2000-0.2

00.20.40.6

500 1000 1500 2000

-0.4

-0.2

0

500 1000 1500 2000

-0.1

-0.05

0

500 1000 1500 2000

00.020.040.06

500 1000 1500 2000

a 6mea

n

energy (MeV)

a 4mea

n

energy (MeV)

a 2mea

n

energy (MeV)

a 0mea

n

energy (MeV)

a 5mea

n

energy (MeV)a 3m

ean

energy (MeV)

a 1mea

n

energy (MeV)

Fig. 12. Third-level fits parametrizing the energy dependence of the second-levelparameters describing the Gaussian mean and width parameters.

Gaussian parameters on µdirectCh at E0. These are evaluated at the particular

value of µdirectCh to determine the appropriate mean and width of the Gaussian

distribution which describes the Cherenkov primitive distribution appropriatefor the particular values of µdirect

Ch and E0. The validity of the assumptionsand simplifications can be tested directly by comparing the calculated primi-tive distributions with those actually observed in the Monte Carlo simulatedevents. Figure 13 shows such a comparison.

For the scintillation (late) primitive distributions, events are generatedwith scintillation light only, and the tc histograms are created in bins of µdirect

sci .Each tc histogram is fit to a sum of two exponentials convolved with a Gaussiandistribution. The exponential decay constants are fixed at τ1 = 5 ns and τ2 =30 ns, leaving three free parameters: the time origin, the Gaussian resolution,and the relative weight of the two exponentials. At a given E0, the threeparameters are extracted in each histogram to obtain their dependence onµdirect

sci , which is parametrized as a sixth-order polynomial in log(µdirectsci ). The

dependence of the three sets of seven parameters from the polynomials areparametrized as a function of E0 by fourth-order polynomials, in analogy withthe Cherenkov case. We now have Gsci(t

c;E0, µdirectsci ), the scintillation primitive

20

E0=250 MeV, muons, old tubes, Cherenkov light

corrected time (ns)

dp(t

)/dt

(ns

-1)

0

0.05

0.1

0.15

0.2

0.25

-20 0 20µ=0.05

0

0.05

0.1

0.15

0.2

0.25

-20 0 20µ=0.11

0

0.05

0.1

0.15

0.2

0.25

-20 0 20µ=0.19

0

0.05

0.1

0.15

0.2

0.25

0.3

-20 0 20µ=0.42

0

0.05

0.1

0.15

0.2

0.25

0.3

-20 0 20µ=0.76

0

0.05

0.1

0.15

0.2

0.25

0.3

-20 0 20µ=1.64

00.05

0.10.15

0.20.25

0.30.35

-20 0 20µ=3

00.05

0.10.15

0.20.25

0.30.35

0.40.45

-20 0 20µ=6.42

E0=600 MeV, muons, old tubes, Cherenkov light

corrected time (ns)

dp(t

)/dt

(ns

-1)

00.020.040.060.08

0.10.120.140.16

-20 0 20µ=0.05

00.020.040.060.08

0.10.120.140.160.18

-20 0 20µ=0.11

00.025

0.050.075

0.10.125

0.150.175

0.2

-20 0 20µ=0.19

0

0.05

0.1

0.15

0.2

0.25

-20 0 20µ=0.42

0

0.05

0.1

0.15

0.2

0.25

-20 0 20µ=0.76

0

0.05

0.1

0.15

0.2

0.25

0.3

-20 0 20µ=1.64

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-20 0 20µ=3

00.05

0.10.15

0.20.25

0.30.35

0.4

-20 0 20µ=6.42

E0=1500 MeV, muons, old tubes, Cherenkov light

corrected time (ns)

dp(t

)/dt

(ns

-1)

0

0.02

0.04

0.06

0.08

0.1

0.12

-20 0 20µ=0.05

00.010.020.030.040.050.060.070.080.09

-20 0 20µ=0.11

0

0.02

0.04

0.06

0.08

0.1

-20 0 20µ=0.19

0

0.02

0.04

0.06

0.08

0.1

-20 0 20µ=0.42

00.020.040.060.08

0.10.120.14

-20 0 20µ=0.76

00.020.040.060.08

0.10.120.140.160.18

-20 0 20µ=1.64

0

0.05

0.1

0.15

0.2

0.25

-20 0 20µ=3

0

0.05

0.1

0.15

0.2

0.25

-20 0 20µ=6.42

Fig. 13. Check of primitive distributions. The parametrized tc likelihood distri-butions for Cherenkov light are compared against the actual tc distributions for250 (top), 600 (middle) and 1500 (bottom) MeV muons as a function of predictedcharge. (Note that the high-E0, low-µ hits, which are less well modeled, are rare.)

distribution. Figure 14 shows the primitive distributions compared with actualtc distributions from the Monte Carlo simulation for muons at E0 = 250, 600and 1500 MeV.

21

E0=250 MeV, muons, old tubes, scintillation light

corrected time (ns)

dp(t

)/dt

(ns

-1)

0

0.01

0.02

0.03

0.04

0.05

0 50µ=0.04

0

0.01

0.02

0.03

0.04

0.05

0 50µ=0.09

0

0.01

0.02

0.03

0.04

0.05

0 50µ=0.16

0

0.01

0.02

0.03

0.04

0.05

0 50µ=0.33

0

0.01

0.02

0.03

0.04

0.05

0.06

0 50µ=0.58

00.010.020.030.040.050.060.070.08

0 50µ=1.17

00.010.020.030.040.050.060.070.080.09

0 50µ=2.06

00.020.040.060.08

0.10.120.14

0 50µ=4.2

E0=600 MeV, muons, old tubes, scintillation light

corrected time (ns)

dp(t

)/dt

(ns

-1)

00.005

0.010.015

0.020.025

0.030.035

0 50µ=0.04

00.005

0.010.015

0.020.025

0.030.035

0 50µ=0.09

00.005

0.010.015

0.020.025

0.030.035

0 50µ=0.16

00.005

0.010.015

0.020.025

0.030.035

0.04

0 50µ=0.33

00.005

0.010.015

0.020.025

0.030.035

0.040.045

0 50µ=0.58

0

0.01

0.02

0.03

0.04

0.05

0 50µ=1.17

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 50µ=2.06

0

0.02

0.04

0.06

0.08

0.1

0 50µ=4.2

E0=1500 MeV, muons, old tubes, scintillation light

corrected time (ns)

dp(t

)/dt

(ns

-1)

0

0.005

0.01

0.015

0.02

0.025

0 50µ=0.04

0

0.005

0.01

0.015

0.02

0.025

0 50µ=0.09

0

0.005

0.01

0.015

0.02

0.025

0 50µ=0.16

0

0.005

0.01

0.015

0.02

0.025

0.03

0 50µ=0.33

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 50µ=0.58

00.005

0.010.015

0.020.025

0.030.035

0.040.045

0 50µ=1.17

0

0.01

0.02

0.03

0.04

0.05

0 50µ=2.06

00.010.020.030.040.050.060.070.08

0 50µ=4.2

Fig. 14. Check of scintillation primitive distributions. The parametrized tc likelihooddistributions for scintillation light are compared against the actual tc distributionsfor 250 (top), 600 (middle) and 1500 (bottom) MeV muons as a function of predictedcharge.

To obtain the complete PDF ftc(tc) for a given PMT hit, we first divide the

predicted charge into “prompt” and “late” components based on the sources:

µprompt = 0.95µdirectCh

µlate = 0.05µdirectCh + µindirect

Ch + µdirectsci + µindirect

sci

(26)

22

These definitions of µprompt and µlate encapsulate the following assump-tions. First, all prompt light is due to direct Cherenkov light; all other light(including indirect Cherenkov light) follows the late scintillation-based timedistribution. The 5% of direct Cherenkov light which is included in µlate ac-counts for the late pulsing of PMTs which causes promptly arriving light toappear at late times. The late time distribution is used to model indirectCherenkov light since the fluorescence, scattering, and reflection processeswhich lead to indirect light have a time structure similar to that of scintil-lation. Further, late timing is less critical to the reconstruction, as promptlight provides essentially all the useful timing information.

The µprompt and µlate values are used to combine the prompt and lateprimitive distributions to form the total PDF ftc(t

c). In the process, one mustaccount for the fact that a PMT can register only one hit for a given trackdue to the latency period of the electronics. Since the time reported by thePMT reflects the time of the first photoelectron, we assume that a PMT hitthat includes a prompt photoelectron obeys the prompt primitive distributionregardless of how many late photoelectrons may follow. The Poisson distri-bution gives us the probabilities that no prompt or late photoelectrons areproduced based on their expected mean values:

P (no prompt PEs) = e−µprompt

P (no late PEs) = e−µlate

(27)

With these probabilities, one can calculate the probability that a given hitcontains at least one prompt photoelectron:

P (prompt PE present|hit) =1− P (no prompt PEs)

1− P (no prompt PEs)P (no late PEs)(28)

We assign this as the weight wp for the prompt primitive distribution withinthe total PDF, while wl = 1− wp is used as the weight for the late primitivedistribution, yielding:

ftc(tc;E0, µprompt, µlate) = wpGCh(tc;E0, µprompt) + wlGsci(t

c;E0, µlate)(29)

where GCh and Gsci are the prompt Cherenkov-based and late scintillation-based timing distributions, respectively. Note that Eq. (28) removes the overallprobability of a hit occurring. Thus, even if the absolute probability of aprompt photoelectron is small, wp ∼ 1 if µlate µprompt.

23

Fig. 15. Internal fit parameters for (a) a single muon or electron track and (b) twophoton tracks. Each photon track includes a conversion distance parameter s. Thetwo-track parameters can be constrained such that the invariant mass of the twophotons Mγγ is always Mπ0 .

5 The Two-track Reconstruction

The single track model with seven parameters is sufficient for reconstruct-ing νµ and νe CCQE events as well as cosmic muons or their decay electrons.As discussed in Section 2, NC π0 events require a two-track model. Figure15 shows the 12 parameters needed to describe two photon tracks originatingfrom a common vertex. The electron track model serves double-duty as thephoton track model, under the assumption that a showering electron is indis-tinguishable from a showering photon apart from the conversion distance. (Themean conversion length in Marcol 7 is 67 cm.) While the two-track model isconceptually a straightforward extension of the one-track case, complexities ofthe likelihood space require special treatment in the likelihood maximizationalgorithm.

5.1 Two-track Charge Likelihood

Since the charge likelihood Fq depends only on the measured charge q andthe total predicted charge µ at each PMT, the two-track charge likelihood canbe obtained by adding together the predicted charges from the two tracks toobtain the total predicted charge at each PMT:

µ = µtrack 1 + µtrack 2 (30)

24

5.2 Two-track time likelihood

The one-track time likelihoods account for the “first photoelectron only”nature of the electronics by calculating the probability that a prompt photo-electron is present and by then weighting the prompt primitive distribution’sinfluence on the full time PDF accordingly. This scheme is extended to handletwo tracks as follows.

The two single-track primitive distributions, GiCh and Gi

sci are formed foreach track, where i labels the track number. In anticipation of aggregating alllate light later, the late “scintillation” primitive distributions are averaged toform:

Gsci =1

2

(G1

sci +G2sci

)(31)

Of the two tracks, one will have a midpoint that is nearer to the target PMTthan the other. This track’s quantities are labeled with “n” (near); the other’s,“f” (far). In analogy with Eq. (26), we define:

µprompt,n ≡ 0.95 µdirectCh,n

µprompt,f ≡ 0.95 µdirectCh,f

µlate ≡ µtot − µprompt,n − µprompt,f

(32)

The probabilities of not obtaining a photoelectron from these sources are:

P n ≡ P (no prompt PE from near track) = e−µprompt,n

P f ≡ P (no prompt PE from far track) = e−µprompt,f

P l ≡ P (no late PEs) = e−µlate

(33)

from which the following weights are obtained:

wn =1− P n

1− P nP fP l

wf =1− P f

1− P fP l

(1− wn)

wl = 1− wn − wl

(34)

25

where wn is the probability that a prompt photoelectron from the near trackexists given that any photoelectron exists, etc. The weights are used to com-bine the two prompt primitive distributions and the averaged late primitivedistribution:

ftc(tc) = wn GCh,n(tc) + wf GCh,f(t

c) + wl Gsci(tc) (35)

resulting in the complete two-track corrected time PDF.

6 Minimization of − log(L)

6.1 One-track

With the likelihood L defined, the parameter set x that minimizes its neg-ative logarithm F ≡ − log(L) must be found. In the single track minimizationprocess, two issues arise.

First, the energy parameter E0 is tied to the geometry of the event viathe track profiles ρ(s;E0). If the spatial parameters (x0, etc) are varied si-multaneously with E0, the minimization algorithm can get confused by thiscorrelation. The solution is to iterate the minimization process, as describedbelow.

A second issue arises from the discrete PMT lattice, which imprints asmall fluctuating signal on L(x) despite the smooth parameterizations of theinput tables. Minimization algorithms that rely on gradients or that do nothave controllable step sizes are thus poorly suited to the problem. Therefore,the SIMPLEX method in Minuit[11] is chosen over the MIGRAD method.

To initiate the minimization process, a vector of seed parameters is ob-tained from a fast fitter. In the first iteration of the minimization, the energyE0 is held at its seeded value while the six remaining parameters (x0, y0, z0, t0,θ0, φ0) are varied via Minuit/SIMPLEX to find a temporary minimum of F .These six parameters are then fixed, while E0 is freed for a second iteration.Finally, E0 is fixed once more with the six other parameters varied (as in theinitial iteration), to find the final minimum of F. The parameters from thefinal SIMPLEX call are returned as the best-fit set x.

26

Fig. 16. The two π0 fitter configurations discussed in the text.

6.2 Two-track

Figure 16 shows two situations that may arise in fitting a π0 event. Theevent has two photon tracks whose Cherenkov rings are represented in black,while green and red represent two ring configurations corresponding to dif-ferent starting values for the parameters. In case (a), the 12 parameters arenear the correct configuration. In case (b), the parameters are such that bothtracks are directed toward the dominant ring. The latter case corresponds to alocal minimum which may offer a worse likelihood than case (a); however, theminimization algorithm may be trapped by the fact that any small changesto the parameters result in an increased value of F . For example, sweepingone track over to the smaller black ring involves passing through a region withlittle detected Cherenkov light. The intermediate states form a barrier of dis-favored values of F that the minimization algorithm code will have difficultyovercoming, especially if several parameters must be adjusted simultaneouslyeven to make the distant configuration an improvement.

The two-track fits require a minimization approach that avoids these trapsin the likelihood surface. Monte Carlo π0 events were used to identify bothcommon and rare trapping scenarios, and the minimization algorithm ad-dresses each of these through the following collection of seed configurationsand minimization sequences:n

27

• The conversion lengths s1 and s2 are seeded with either 50 cm or 250 cm,leading to four possible pairs.• θ1 and φ1 are seeded with the results from the electron single-track fit or

with one of the eight perturbations (see below).• θ2 and φ2 are seeded with the “best” directions from a full grid of tested

directions (see below).• The four-vertex of the event is seeded with the four-vertex from the electron

single-track fit shifted according to s1, θ1 and φ1.• E1 is seeded with approximately E0 from the electron single-track fit, while

the seed for E2 is, when possible, the energy needed to give an invariantmass equal to Mπ0 .

For the last item, the energy seeds are based on an empirical expressionthat accounts for the second ring’s energy contribution to the one-track fitenergy as a function of the angle between the two γ tracks. The function isobtained by simulating NC π0 events and reconstructing them with the singletrack electron fit. Using the true photon energies (E1 > E2) and opening angleθγγ, the relationship between (E0 − E1)/E2 and cos θγγ is parametrized by alinear function. This function is used to determine the seed E1 and E2 valuesfor a given E0 and cos θγγ.

The nine (1+8) possible seed directions for track 1 are created as follows.The spatial distribution of PMT charge is projected onto a plane perpendicu-lar to the electron fit direction. The major axis of the covariance ellipse of theresulting two-dimensional distribution is found. The nine possible seed direc-tions are rotations of the best electron-fit direction by 0, ±0.159, ±0.450 and±0.644 radians parallel to the major axis and ±0.159 radians perpendicularto the major axis.

The seed directions for track 2 come from either a “coarse” 24×12 (θ, φ)grid or a “fine” 50×25 (θ, φ) grid. The grid type is specified at run time.

Of the 9×(24×12)=2592 or 9×(50×25)=11250 possible combinations oftrack 1 and track 2 seed directions, only six are used to seed actual minimiza-tion sequences. They are the six combinations with (1) the best total likelihood(time and charge), (2) the best charge likelihood when the estimated track en-ergies 3 are similar (0.5 < E1

E2< 2), (3) the best charge likelihood when the

estimated track energies are dissimilar, and (4)−(6) the second-best likelihoodin each of these three cases. A small exception: combinations involving one ofthe eight perturbed track 1 directions and having estimated track energies thatdiffer by more than a factor of five are not considered. In these energeticallyasymmetric cases, the electron fit direction would be a good representation oftrack 1 since the influence of track 2 would be small, so there is no reason to

3 Recall the energy seed procedure discussed above.

28

consider alternate track 1 directions.

The six best and second-best track 1 and track 2 direction and energycombinations are chosen anew for each permutation of (s1, s2), leaving 24complete parameter sets in total. Each set seeds two Minuit sequences. Insequence 1, the SIMPLEX minimization routine is first run with the energyand angle parameters held fixed; once a minimum is found, SIMPLEX is calledagain with all parameters free. In sequence 2, only the energy parametersare held fixed in the initial SIMPLEX call. This results in 48 fit sequences,each of which ends with a completely free SIMPLEX minimization. The bestparameter set (i.e., the one that results in the lowest F ) from the forty-eightfinal SIMPLEX calls is reported as the best parameter set x for the two-trackfit. During all minimizations, the photon conversion points are constrained tobe within the main detector volume. This prevents the fit from removing theinfluence of a track through an inflated conversion length. The event vertexitself, however, is not constrained.

The SIMPLEX minimizations can be performed with a constraint on theinvariant mass. This is accomplished by removing E2 as a free parameter andsetting it via the relation:

E2 =M2

π0

2E1(1− cos θγγ)(36)

where θγγ is the angle between the photon tracks. This fixed-mass mode is theactual π0 hypothesis, while the free-mass mode allows for mass reconstruction.Both are used for π0 identification, with the former lending its maximizedlikelihood Lπ0 and the latter providing a reconstructed mass Mγγ.

7 Performance

The performance of the reconstruction algorithm on simulated neutrinointeractions (νµ CCQE, νe CCQE and NC π0 events) generated with NU-ANCE [8] and propagated through the detector Monte Carlo simulation isshown in Figures 17−21. 4 The electron and muon one-track fits are appliedto the νµ and νe CCQE events, respectively, while the two-track fits, withand without the mass constraint, are applied to the NC π0 events. A minimalselection is applied to ensure that the events are contained within the main

4 For comparisons of Monte Carlo to data and for discussion of the developmentand validation of the Monte Carlo, see Refs. [9] and [10].

29

detector region and, for plots not directly related to vertex reconstruction,within the fiducial volume of the analysis.

A few observations about the figures, particularly Figure 18: (1) The reso-lutions for νµ CCQE events degrade as the energy approaches the Cherenkovthreshold from above, as most of the information in the event comes from the(rapidly decreasing) prompt Cherenkov light. (2) A low-energy π0 offers littleinformation on its direction, as all that is observed is the isotropically producedγγ final state. Thus, the angular resolution (but not the vertex resolution) be-comes poor. (3) The vertex resolution for π0 events is worse than νe CCQEevents since the former has spatially displaced light production thanks to the∼70 cm average photon conversion length. (4) Typical performance numbersfor νe CCQE events: 20 cm vertex resolution, 12% kinetic energy resolution,and 4 angular resolution. (5) Typical performance numbers for νµ CCQEevents: 10 cm vertex resolution, 8% kinetic energy resolution and 2 angularresolution. (6) Figure 19 shows a π0 mass resolution of ∼13%.

0

0.1

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1

true R (cm)0 100 200 300 400 500 600

fitt

ed R

(cm

)

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600

fid

uci

al v

olu

me

cut

PM

T f

aces

op

tica

l bar

rier

CC QEµν

0

0.1

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1

true KE (MeV)0 200 400 600 800 1000 1200 1400

fitt

ed K

E (

MeV

)

0

200

400

600

800

1000

1200

1400 CC QEµν

0

0.1

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true R (cm)0 100 200 300 400 500 600

fitt

ed R

(cm

)

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600

fid

uci

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me

cut

PM

T f

aces

op

tica

l bar

rier

CC QEeν

0

0.1

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true KE (MeV)0 200 400 600 800 1000 1200 1400

fitt

ed K

E (

MeV

)

0

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400

600

800

1000

1200

1400 CC QEeν

0

0.1

0.2

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true R (cm)0 100 200 300 400 500 600

fitt

ed R

(cm

)

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600

fid

uci

al v

olu

me

cut

PM

T f

aces

op

tica

l bar

rier

0πNC

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

true KE (MeV)0 200 400 600 800 1000 1200 1400

fitt

ed K

E (

MeV

)

0

200

400

600

800

1000

1200

1400

0πNC

Fig. 17. Reconstructed radial event position (left) and kinetic energy (right) plottedagainst their true values. The top, middle, and bottom panels show νµ CCQE, νeCCQE, and NC π0 events.

30

true KE (MeV)0 200 400 600 800 1000 1200

R r

eso

luti

on

(cm

)

0

10

20

30

40

50

60 CC QEµν CC QEeν

e0π NC

true KE (MeV)0 200 400 600 800 1000 1200

frac

tio

nal

KE

res

olu

tio

n

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50 CC QEµν CC QEeν

e0π NC

true KE (MeV)0 200 400 600 800 1000 1200

aver

age

dir

ecti

on

err

or

(deg

rees

)

0

2

4

6

8

10

12

14 CC QEµν CC QEeν

e0π NC

Fig. 18. Radius, kinetic energy, and direction resolutions. The jitter is due to limitedtest sample statistics in some regions. The νe curves cut off at 100 MeV due to anunrelated upstream cut on the Monte Carlo sample.

31

)2fitted invariant mass (MeV/c0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08 < 70 MeV0π: 0 MeV < KE0πNC

2peak mass: 151 MeV/c

resolution: 12%

)2fitted invariant mass (MeV/c0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

< 525 MeV0π: 350 MeV < KE0πNC

2peak mass: 132 MeV/c

resolution: 13%

)2fitted invariant mass (MeV/c0 50 100 150 200 250 3000.00

0.02

0.04

0.06

0.08

0.10

< 175 MeV0π: 70 MeV < KE0πNC

2peak mass: 143 MeV/c

resolution: 11%

)2fitted invariant mass (MeV/c0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

< 875 MeV0π: 525 MeV < KE0πNC

2peak mass: 129 MeV/c

resolution: 17%

)2fitted invariant mass (MeV/c0 50 100 150 200 250 3000.00

0.02

0.04

0.06

0.08

0.10 < 350 MeV0π: 175 MeV < KE0πNC

2peak mass: 136 MeV/c

resolution: 13%

)2fitted invariant mass (MeV/c0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

0.05

< 1400 MeV0π: 875 MeV < KE0πNC

2peak mass: 133 MeV/c

resolution: 28%

Fig. 19. Fitted invariant mass Mγγ for Monte Carlo-simulated NC π0 events in threelow energy regions (left) and three high energy regions (right).

0

0.1

0.2

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0.5

0.6

0.7

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0.9

1

(cm)largertrue E0 200 400 600 800 1000 1200 1400

(cm

)as

soc

fitt

ed E

0

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800

1000

1200

1400

0πNC

0

0.1

0.2

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(cm)smallertrue E0 100 200 300 400 500 600 700

(cm

)as

soc

fitt

ed E

0

100

200

300

400

500

600

700

0πNC

Fig. 20. Reconstructed γ energies in Monte Carlo-simulated NC π0 events plottedagainst true values. The left (right) panel shows the higher (lower) energy γ fromeach event. The association of each fitted track to the underlying true γ’s is, ingeneral, ambiguous. For these plots, we choose the assignment that gives the smallercombined energy and direction discrepancy.

32

(MeV)larger/smallertrue E0 200 400 600 800 1000

res

olu

tio

nas

soc

frac

tio

nal

E

0.0

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0.5

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0.7

0.8

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1.00πNC

γ higher energy

γ lower energy

(MeV)larger/smallertrue E0 200 400 600 800 1000

aver

age

dir

ecti

on

err

or

(deg

rees

)

0

2

4

6

8

10

12

14

0πNC

γ higher energy

γ lower energy

Fig. 21. Energy (left) and direction (right) resolutions for γ’s in simulated NC π0

events. For a given true γ energy, a dominant γ has expectedly better resolutionthan does a subordinate γ.

8 Particle Identification

The maximum likelihoods returned from each fit can be used for hypothesistesting. The two quantities

Re/µ ≡ logLeLµ

= logLe − logLµ (37)

and

Re/π ≡ logLeLπ

= logLe − logLπ (38)

are used to determine for a given event whether the electron hypothesis ispreferred over the muon and π0 hypotheses. In these expressions, Le, Lµ andLπ are the maximized likelihoods returned by the electron, muon, and (fixed-mass) two-track fits, respectively.

8.1 Electron/Muon separation

Figure 22 shows for simulated νe CCQE events and νµ CCQE events thedistribution of Re/µ as a function of the energy reconstructed by the electronfit. For each sample, the events have been subject to a set of preselectioncriteria used in the νe appearance analysis. These require that there is only onetime-cluster of PMT hits in the event, eliminating obvious νµ CC events thatproduce decay electrons. Cosmic backgrounds are eliminated by requiring theminimum number of PMT hits in the main region to be >200 and the numberof veto hits to be <6. Furthermore, the average time of hits is required to liewithin the expected beam delivery window from the Booster.

33

Energy (GeV)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

) µ/L e

log

(L

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

CCQEeν

Energy (GeV)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

) µ/L e

log

(L

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

CCQEµν

Fig. 22. Distribution of Re/µ= log(Le/Lµ) for Monte Carlo simulated νe CCQEevents (top) and νµ CCQE events (bottom) as a function of reconstructed energy(from the electron hypothesis fit).

34

Energy (GeV)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) π/L e

log

(L

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1 CCQEeν

Energy (GeV)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) π/L e

log

(L

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.10πNC

Energy (GeV)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

)2 (

MeV

/cγγ

M

0

100

200

300

400

500

600

700

800

900

1000

CCQEeν

Energy (GeV)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

)2 (

MeV

/cγγ

M

0

100

200

300

400

500

600

700

800

900

1000

0πNC

Fig. 23. Left: Distribution ofRe/π= log(Le/Lπ) for Monte Carlo simulated νe CCQEevents (top) and NC π0 events (bottom). Right: same for Mγγ . The black lines oneach distribution show the selection used in the MiniBooNE νµ → νe oscillationanalysis [4].

For the νe CCQE events, Re/µ tends to take positive values, indicatingthat the fit to the event with the electron hypothesis is favored over the muonhypothesis. Likewise, Re/µ tends to be negative for the νµ CCQE events, in-dicating that the muon hypothesis fits these events better than the electronhypothesis. At high energies (>1 GeV), the electron/muon separation is aidedby the fact that the muon pathlength grows approximately linearly with en-ergy, while the electron shower grows more slowly.

8.2 Electron/π0 separation

The left plots in Figure 23 show the distributions of Re/π as a function ofenergy reconstructed by the single-track electron reconstruction for simulatedνe CCQE events and NC π0 events. The quantity Re/π tests whether a givenevent fits better as a single electron track or as a π0. Overall, electron/π0

separation becomes more difficult at high energies as the energies of the decayphotons in π0 events become more asymmetric and the shower fluctuationsin a single electron event get large enough to mimic the presence of a secondphoton. Also, π0 events in which one of the two photons leaves the detectorunconverted present an essentially irreducible background at low energy.

35

Another quantity that can be used for electron/π0 separation is Mγγ, theinvariant mass of the two photons returned by the free-mass two-track fit. TheNC π0 events peak at the π0 mass, whereas the νe CCQE events peak towardzero. This is seen in the right plots in Figure 23, where the Mγγ distribution isshown as a function of reconstructed energy. The use of the true π0 mass in thefit’s seeding procedure (Section 6.2) induces no appreciable high-mass peak inthe νe CCQE events while helping considerably in the correct identification ofNC π0 events.

The νe appearance analysis at MiniBooNE utilizes both Mγγ and Re/π toseparate νe CCQE events from NC π0.

8.3 Particle Identification Performance

The background rejection levels reachable with these fitter-derived quan-tities are shown for Monte Carlo events in Figures 24 and 25. Figure 24 showsthe efficiency, after fiducial volume and cosmic rejection cuts, for selectingsignal νe CCQE events alongside the efficiency (or misidentification rate) forνµ CCQE events. Three prototype selections are used, each chosen to give anapproximately fixed signal efficiency at all energies. Note that the misidentifi-cation rate is shown with a logarithmic vertical scale. Figure 25 shows similarplots for π0 rejection, one for the Re/π cut and one for the Mγγ cut.

Energy (GeV)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Eff

icie

ncy

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1) Selectionµ/L

elog(L

Selection 1

Selection 2

Selection 3

Energy (GeV)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Mis

iden

tifi

cati

on

Rat

e

−310

−210

−110) Selectionµ/L

elog(L

Selection 1

Selection 2

Selection 3

Fig. 24. Performance of three prototype Re/µ= log(Le/Lµ) cuts for the νe selection.Left: Signal (νe CCQE) efficiency versus the electron reconstruction’s energy. Right:Misidentification rate of νµ CCQE events versus the electron reconstruction’s energy.

Misidentification rates of a few percent for νµ CCQE events are seen for sig-nal efficiencies from 70−90%. (Note that νµ CCQE events are further reducedby about an order of magnitude by recognizing the delayed muon decay.) Thetwo π0 variables indicate that the misidentification rate increases with energy;

36

E (GeV)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Eff

icie

ncy

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75) Selectionπ/L

elog(L

Selection 1

Selection 2

Selection 3

E (GeV)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Mis

iden

tifi

cati

on

Rat

e

−210

−110

) Selectionπ/Le

log(L

Selection 1

Selection 2

Selection 3

E (GeV)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Eff

icie

ncy

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8 SelectionγγM

Selection 1

Selection 2

Selection 3

E (GeV)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Mis

iden

tifi

cati

on

Rat

e

−210

−110

SelectionγγM

Selection 1

Selection 2

Selection 3

Fig. 25. Performance of three prototype Re/π= log(Le/Lπ) (top) and Mγγ (bot-tom) cuts for the νe selection. Left: Signal (νe CCQE) efficiency versus the electronreconstruction’s energy. Right: Misidentification rate of NC π0 events versus theelectron reconstruction’s energy.

this is expected, since the two decay photons become highly asymmetric ener-getically and since the Cherenkov rings can achieve a greater degree of overlap.The rise in the π0 misidentification rate at low energies is due to events withan escaping photon (corresponding to the cluster of π0 events with positiveRe/π at 50−200 MeV in Figure 23). Between 200 MeV and 600 MeV, wheremost of the NC π0 events lie, a misidentification rate of less than 1 percent canbe achieved with 40% efficiency using only the Re/π cut, while rates of a fewpercent can be achieved with only the Mγγ cuts. In the MiniBooNE νµ → νeoscillation analysis, 50% signal efficiency and 1% π0 misidentification rate areobtained by requiring both Mγγ < Mmax

γγ (E) and Re/π > Rmaxe/π (E), where the

“max” functions are quadratic in the energy E obtained from the single-trackelectron fit. These quadratic selection boundaries are indicated by the solidblack lines in Figure 23. These cuts, along with an analogous Re/µ cut, weretuned to optimize sensitivity to LSND-like oscillations.

37

9 Summary

A maximum likelihood reconstruction algorithm is used at MiniBooNE todetermine track parameters under electron, muon, and π0 hypotheses. Under-lying the likelihood is a track and detector model that calculates for a givenparameter set the charge and time PDFs expected for each PMT, accountingfor the spatially extended production of Cherenkov and scintillation light aswell as the effects of indirect light from subsequent optical processes.

The maximized likelihoods for a given event obtained under different eventhypotheses are used for event selection. In particular, the ratio of the likeli-hoods under the electron and muon models is used to suppress νµ CC events inthe MiniBooNE νµ → νe oscillation search; the ratio of the likelihoods underthe electron and two-track π0 fit (with fixed invariant mass) and the invariantmass obtained from the free-mass two-track fit are used to suppress NC π0

events. While the reconstruction has been developed within the context of theMiniBooNE νµ → νe oscillation search, other experiments employing similarCherenkov detection techniques should be able to use the techniques discussedhere.

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